If we give the name of virtual momentum (or simply of momentum in its primitive sense) to the product of each force by its virtual velocity, — a product which in fact then measures the effort of the force to move the machine, — we may greatly simplify the statement of the principle in merely saying that, in this case, the momentum of the two forces must be equal and of opposite signs, that there may be equilibrium, and that the positive or negative sign of each momentum is determined according to that of the virtual velocity, which will be considered positive or negative according as, by the supposed motion, the projection of the point of application would be found to fall upon the direction of the force or upon its prolongation. This abridged expression of the principle of virtual velocities is especially iiseful for the statement of this princijjle in a general manner, with regard to any system of forces whatever. It is simply this: that the algebraic si;m of the virtual moments of all forces, estimated according to the jpreced- VIRTUAL VELOCITIES. 129 iug rule, must be null to cause equilibrium: and this condition must exist distinctly witli regard to all the elementary motions which the system might assume in virtue of the forces by which it is aniinated. In the equation, containing this principle, furnished by Lagrange, the whole of Rational Mechanics may be considered to be implicitly comprehended.
While the theorem of vii'tual velocities was conceived of only as a general property of equilibrium, it could be verified by observing its constant conformity with the ordinary laws of equilibrium, otherwise obtained, of which it was a summary, useful by its simplicity and uniformity. But, if it was to be a fundamental principle, a basis of the whole science, it must be underived, or at least caj^able of being presented in its preliminary propositions as a matter of observation. This was done by Lagrange, by his ingenious demonstration through a system of pulleys. He exhibited the theorem of virtual velocities very easily by imagining a single weight which, by means of pulleys suitably constructed, replaces simultaneously all the forces of the system. Many other demonstrations have been furnished; but, while more com^^licated, they are not logically superior.
From the philosophical jjoiut of view it is clear that this general theorem, being a necessary consequence of the fundamental laws of motion, can be deduced in various ways, and becomes practically the point of departure of the whole of Rational Mechanics. A perfect unity having been established by this principle, we need not look for any others; and we may rest assured that Lagrange has carried the co-ordination of the science as far as it can go. The only possible object would be to simplify the analytical researches to which the science is now reduced; and nothinsr can be conceived more admirable for this purpose than Lagrange's adaptation of the ])rinciple of virtual velocities to the uniform application of mathematical analysis.
Striking as is the philosophical eminence of this principle, there are difficulties enough in its use to prevent its being considered elementary, so far as to j^reclude the consideration of any other in a course of dogmatic teaching. It is for this reason that I have referred to the dynamic method, properly so called, which is the only one in general I. K use at present. All other considerations must however be only provisional. Lagrange's method is at pi'esenttoo new; but it is impossible that it should for ever remain in the hands of a small number of geometers, who alone shall be able to make use of its admirable properties. It must become as popular in the mathematical world as the great geometrical conception of Descartes: and this general progress would be almost accomplished if the fundamental ideas of transcendental analysis were as widely spread as they ought to be.
The greatest acquisition, since the regene- Counles ration of the science by Lagrange, is the conception of M. Poinsot, — the theory of Couples, which appears to me to be far from being sufficiently valued by the greater number of geometers. These Couples, or systems of parallel forces, equal and contrary, had been merely remarked before the time of M. Poinsot, as a sort of paradox in Statics. He seized upon this idea, and made it the subject of an extended and original theory relating to the transformation, composition, and use of these singular groups, which he has shown to be endowed with properties remarkable for their generality and simplicity. He used the dynamic method in his study of the conditions of equilibrium: but he presented it, by the aid of his theory of couples, in a new and simplified aspect. But his conception will do more for dynamics than for statics; and it has hardly yet entered upon its chief office. Its value will be appreciated when it is fonnd to render the notion of the movements of rotation as natural, as familiar, and almost as simple as that of forward movement or translation. Share of eqiia- ^^^^ more consideration should, I think, be tions in pro- adverted to before we quit the subject of ducing equili- statics as a whole. When we study the nature bruuii. q£ ^j-^g equations which express the conditions of equilibrium of any system of forces, it seems to me not enough to establish that the sum of these equations is indispensable for equilibrium. I think the fiu'ther statement is necessary, — in what degree each contributes to the result. It is clear that each equation must destroy some one of the possible motions that the body would make in virtue of existing forces; so that the whole of the equations must EQUATIONS OF EQUILIBRIUM. 131 produce equilibrium by leaving an impossibility for the body to move in any way whatever. Now the natural state of things is for movement to consist of rotation and translation. Either of tliese may exist without the other; Ijut the cases ai-e so extremely rare of their being found apart, that the verification of either is regarded by geometers as the strongest presumption of the existence of the other. Thus, when the rotation of the sun uj^on its axis was established, every geometer concluded that it had also a progressive motion, carrying all its planets Avith it, before astronomers had produced any evidence that such was actually the case. In the same way we conclude that certain planets, travelling in their orbits, rotate round their axes, though the fact has not yet been verified. Some equations must therefore tend to destroy all progressive motion, and others all motion of rotation. How many equations of each kind must there be?
It is clear that, to keep a body motionless, it must be hindered from moving according to three axes in different planes — commonly supposed to be perpendicular to each other. If a body cannot move from north to south, nor the reverse; nor from east to west, nor the reverse; nor up, nor down, it is clear that it cannot move at all. Movement in any intermediate direction might be conceived of as partial progression in one of these, and is therefore impossible. On the other hand, we cannot reckon fewer than three independent elementary motions; for the body might move in the direction of one of the axes, without having any translation in the direction of either of the others. Thus we see that, in a general way, three equations are necessary, and three are sufficient to establish the absence of translation; each being specially adapted to destroy one of the three progressive motions of which the body is capable. The same view presents itself with regard to the other motion, — of rotation. The mechanical conception is more complicated; but it is true, as in the simpler case, that motion is possible in only three directions, — in three co-ordinated planes, or round three axes. Three equations are necessary and sufficient here also; and thus we have six which are indis2:)ensable and sxifficient to stop all motion whatever.
When, instead of supposing any system of forces whatever as the subject of the question, we particularize any, we get rid of more or fewer possible movements. Havmg excluded these, we may exclude also their corresponding equations, retaining only those which relate to the possible motions that remain. Thus, instead of having to deal with six equations necessary to equilibrium, there may be only three, or two, or even one, which it will be easy enough to obtain in each case. These remarhs may be extended to any restrictions upon motion, whether resulting from the special constitution of the system of forces, or from any other kind of control, affecting the body under notice. If, for instance, the body were fastened to a point, so that it could freely rotate but not advance, three equations would siiffice: and again, if it is fastened to two fixed points, two equations are enough; and even one, if these two fixed points are so placed as to prevent the body from moving on the axis between them. Finally, its being attached to thi*ee fixed points, not in a right line, will prevent its moving at all, and establish equilibrium without any condition, whatever may be the forces of the system. The spirit of this analysis is entirely independent of any method by which the equations of equilibrium will have been obtained: but the different general methods ai'e far from being equally suitable to the application of this rule. The one which is best adapted to it is, undoubtedly, the Statical one, properly so called, founded, as has been shown, on the principle of virtual velocities. In fact, one of the chai*acteristic properties of this principle is the pei-fect precision with which it analyses the phenomena of equilibrium, by distinctly considering each of the elementary motions permitted by the forces of the system, and furnishing immediately an equation of equilibrium specially relating to this motion.
Connection of When we conje to the inquiry how tlie concrete geometers apply the principles of abstract with the ab- Mechanics to the properties of real bodies, we stract question, j^^^gt state that the only complete application yet accomplished is in the question of terrestrial gravity. Now, this is a subject which cannot, logically, be treated under the head of Mechanics, as it belongs to Physics. It CONNECTIOX OF COXCRETE AND ABSTRACT QUESTION. 133 is sufficient to explain that the statical study of terrestrial gravity becomes convertible into that of centres of gravity; and that all confusion between the two dej^artments of research would be avoided if we accustomed ourselves to class the theory of centres of gravity among the questions of pure geometry. In seeking the centre of gravity as (according to the logical denomination of the ancient geoineters) the centre of mean distances, we remove all traces of the mechanical origin of the Cjuestion, and convert it into this jiroblem of general geometry: — Given, any system of points disposed in a determinate way with regard to each other, to find a point whose distance to any plane shall be a mean between the distances of all the given points to the same plane. — The abstraction of all consideration of gravity is an assistance in every way. The simple geometrical idea is precisely what we want in most of the principal theories of Rational Mechanics, and especially when we contemplate the great dynamic properties of the centre of mean distances; in which study the idea of gravity becomes a mere encumbrance and perplexity. It is true that, by proceeding thus, we exclude the question from the domain of Mechanics, to place it in that of Geometry. I should have so classed it but for an unwillingness to break in upon established customs. However it may be as to the matter of arrangement, it is highly important for us not to misapprehend the true nature of the cjuestion. — The integral calculus offers the means of surmounting those difficulties in determining the centre of gravity which are imposed by the conditions of the question. But, the integrations in this case being more complicated than those to which they are analogous, — those of quadratures and cubatures, — their precise solution is, owing to the extreme imperfection of the integral Calcuhis, much more rarely obtained. It is a matter of high importance, however, to be able to introduce the consideration of the centre of gravity into general theories of analytical mechanics.
Such is, then, the relation of terrestrial gravitation to the science of abstract Statics. As for universal gravitation, no complete study has yet been made of it, except in regard to spherical bodies. What we know of the law of 134 POSITIVE PHILOSOPEIY.
gravitation would easily enable us to compute the mutual attraction of all known bodies, if tlie conditions of each body were understood by us; but this is not the case. For instance, we know nothing of the law of density in the interior of the heavenly bodies. It is still true that the primitive theorems of Newton on the attraction of sj^herical bodies are the most useful part of our knowledge in this direction.
Gravity is the only natural force that we are practically concerned with in Rational Statics: and we see, by this, how backward this science is in regard to universal gravitation. As for the exterior general circumstances, such as friction, resistance of media, and the like, which are altogether excluded in the establishment of the rational laws of Mechanics, we can only say that we are absolutely ignorant of the way to introduce them into the fundamental relations afforded by analytical Mechanics, because we have nothing to rely on, in working them, but precarious and inaccurate hypotheses, unfit for scientific use.
. As for the theory of equilibrium in regard of^fluid"^^"^ ^^ fluid bodies, — the application which it remains for us to notice, — those bodies must be regarded as either liquid or gaseous.
-rj 1,,. Hydrostatics may be treated in two ways.
We may seek tlie uiws of the equinbrium 01 flaids, according to statical considerations pro])er to that class of bodies: or we may look for them among the laws which relate to sohds, allowing for the new characteristics residting from fluidity.
The first method, being the easiest, was in early times the only one employed. Till a rather recent time, all geometers employed themselves in proposing statical principles peculiar to fluids; and especially with regard to the grand question of the figure of the earth, on the supposition that it was once fluid. Huyghens first endeavoured to resolve it, taking for his principle of equilibrium the necessary perpendicularity of weight at the free surface of the fluid. Newton's principle was the necessary equality of weight between the two fluid columns going from the centre, — the one to the pole, tlie other to some point of the equator. Bouguer showed that both methods were bad, EQUILIBRIUM OF FLUIDS. 135 because, though each was iucoutestable, the two failed, in many cases, to give the same form to the fluid mass in equilibrium. But he, in his turn, was wrong in believing that the union of the two principles, when they agreed in indicating the same form, was sufficient for equilibrium. It was Clairaut who, in his treatise on the form of the earth, first discovered the true laws of the case, setting out from the evident consideration of the isolated equilibrium of any infinitely small canal; and, tried by this criterion, he showed that the combination required by Bouguer might take place without equilibrium happening. Several great geometers, proceeding on Clairaut' s foundation, have carried on the theory of the equilibrium of fluids a great way. Maclaurin was one of those to whom we owe much; but it was Euler who brought up the subject to its present point, by founding the theory on the principle of equal presstire in all directions. Observation of the statical constitution of fluids indicates this as a general law; and it furnishes the requisite equations with extreme facility.
It was inevitable that the mathematical theory of the equilibrium of fluids should, in the first place, be founded, as we have seen that it was, on statical principles peculiar to this kind of bodies: for, in early days, the y...
characteristic differences between solids and fluids must have appeared too great for any geometer to think of applying to the one the general principles appropriated to the other. But, when the funda- Cases mental laws of hydrostatics were at length obtained, and men's minds were at leisure to estimate the real diversity between the theories of fluids and of solids, they could not but endeavour to attach them to the same general piinciples, and perceive the necessary applicability of the fundamental rules of Statics to the equilibrium of fluids, making allowance for the attendant variability of form. But, before hydrostatics could be comprehended under Statics, it was necessary that the abstract theory of equilibrium should be made so general as to apply directly to fluids as well as solids. This was accomplished when Lagrange supplied, as the basis of the whole of Rational Mechanics, the single principle of Virtual Velocities. One of its most valuable properties is its being as directly applicable to fluids as to solids. From that time, Hydrostatics, ceasing to be a natui'al branch of science, lias taken its place as a secondary division of Statics. This arrangement has not yet been familiarly admitted; but it must soon become so.
To see how the principle of Virtual Velocities may lead to the fundamental equations of the equilibrium of fluids, we have to consider that all that such an ajjplication requires, is to introduce amoug the forces of the system under notice one new force, — the pressure exerted upon each molecule, which will introduce one term more into the general equation. Proceeding thus, the three general equations of the equilibrium of fluids, employed when hydrostatics was treated as a separate branch, will be immediately reached. If the fluid be a liquid, we must have regard to the condition of incompressibility, — of change of form without change of volume. If the fluid be gaseous, we must substitute for the incompressibility that condition which sul)]ects the volume of the fluid to vary according to a determinate function of the pressiire; for instance, in the inverse ratio of the pressure, according to the physical law on which Mariotte has founded the whole Mechanics of the gases. We know but too little yet of these gaseous conditions; for Mariotte's law can at present be regarded only as an approximation, — sufficiently exact for average circumstances, but not to be rigorously applied in any case whatever.
Some confirmation of the philosophical character of this method of treating hydrostatics arises from its enabling us to pass, almost insensibly, from the order of bodies of invariable form to that of the most variable of all, through intermediate classes, — as flexible and elastic bodies, — whereby we obtain, in an analytical view, a natural filiation of subjects.
We have seen how the department of Statics has been raised to that high degree of speculative perfection which transforms its questions into simj^le problems of Mathematical Analysis. We must now take a similar review of the other department of general Mechanics, — that more extended and more complicated study which relates to the laws of Motion.
THEORY OF RECTILINEAR MOTION. 137 DYNAMICS.
The object of Dynamics is the study of the ^1 •, varied motions produced by continuous forces. The Dynamics of varied motions or continuous forces includes two departments, — the motion of a point, and that of a body. From the positive point of view, this means that, in certain cases, all the parts of the body in question have the same motion, so that the determination of one particle serves for the whole; while in the more general case, each particle of the body, or each body of the system, assuming a distinct motion, it is necessary to examine these different effects, and the action upon them of the relations belonging to the system under notice. The second theory being more complicated than the first, the first is the one to begin with, even if both are deduced from the same principles.
With regard to the motion of a point, the question is to determine the circumstances of the compound curvilinear motion, resulting from the simidtaueous action of different continuous forces, it being known what would be the rectilinear motion of the body if influenced by any one of these forces. Like evei-y other, this problem admits of a converse solution.
But here intervenes a preliminary theory, „. j.
which must be noticed before either of the ^^jjj^gj^. ju^^^^j^j^ two departments can be entered upon. This theory is popularly called the tlieory of rectilinear motion, produced by a single continuous force acting indefinitely in the same direction. It may be asked why we want this, after having said that the effect of each separate force is supposed to be known, and the eff'ect of their union the thing to be sought. The answer to this is, that the varied motion produced by each continuous force may be defined in several ways, which depend on each other, and which could never be given simultaneously, though each may be separately the most suitable; whence results the necessity of being able to pass from any one of them to all the rest. The preliminary theory of varied motion relates to these transformations, and is therefore inaptly termed the study of the a.ction of a single force. These different equivalent definitions of the same varied motions result from the simultaneous consider'ation of the three distinct but co related functions which are presented by it, — space, velocity, and force, conceived as dependent on time elapsed. Taking the most extended view, we may say that the definition of a varied motion may be given by any equation containing at once these four variables, of which only one is independent, — time, space, velocity, and force. The problem will consist in deducing from this equation the distinct determination of the three characteristic laws relating to space, velocity, and force, as a function of time, and, consequently, in mutual correlation. This general problem is always reducible to a purely analytical research, by the help of the two dynamical formulas which express, as a function of time, velocity and force, when the law of space is supposed to be known. The infinitesimal method leads to these formulas with the utmost ease, the motion being considered uniform during an infinitely small interval of time, and as uniformly accelerated during two consecutive intervals.
Thence the velocity, supposed to be constant at the instant, according to the first consideration, will be naturally expressed by the diiferential of the space, divided by that of the time; and, in the same way, the continuous force, according to the second consideration, will evidently be measured by the relation between the infinitely small increment of the velocity, and the time employed in producing this increment.
Lagrange's conception of transcendental analysis excluding him from this use of the infiuitesinaal method for the establisment of the two foregoing dynamic formulas, he was led to present this theory under another point of view, more important than seems to be generally supposed. In his Theory of Analytical Functions, he has shown that this dynamic consideration really consists in conceiving any varied motion as compounded, each moment, of a certain uniform motion and another motion uniformly varied, — likening it to the vertical motion of a heavy body under a first impulsion. Lagrange has not given its due advantage to this conception, by developing it as he might THE TWO CASES OF RATIONAL DYNAMICS. 139 have done. In fact, it supplies a complete theory of the assimilation of motions, exactly like the theory of the contacts of curves and surfaces, in the department of geometry. Like that theory, it removes the limits within which we supposed ourselves to be confined, by disclosing to us, in an abstract way, a much more perfect measure of all varied motion than we obtain by the ordinary theory, though reasons of convenience compel us to abide by the method originally adopted.
The first case or department of rational,r.• ^ T • xij.i?xii- c -J. Motion of a dynamics, — that or the motion ot a point, or point of a body which has all its points or portions affected by the same force, — relates to the study of the curvilinear motion produced by the simultaneous action of any different continuous forces. This case divides itself again into two, — according as the mobile point is free, or as it is compelled to move in a single curve, or on a given surface. The fundamental theory of curvilinear motion may be established in either case, in a different way; each being susceptible of direct treatment, and of being connected with the other. In the first case, in order to deduce the second, we have only to regard the active or passive resistance of the prescribed curve or surface as a new force to be added to the others proposed. In the other way, we have only to consider the moving point as compelled to describe the curve which it must traverse; and this is enough to afford the fundamental equations, though this curve may then be primitively unknown.
The other, more real and more difiicult., ^.. J.T, £,\.- £, n Motion 01 a case, is tiiat ot the motion or a system ot sv^teni bodies in any way connected, whose proper motions are altered by the conditions of their connection. There is a new elementary conception about the measurement of forces which some geometers declare to be logically deducible from antecedent considerations, and to which they would assign the place and title of a fourth law of motion. For the sake of convenience we may make it into a fourth law of motion; but such is not its philosophical character. The idea is, that forces which impress the same velocity on different masses are to each other exactly as those masses; or, in other words, that the forces are proportional to the masses, as we have seen them, under the third law of motion, to be proportional to the velocities. All phenomena, such as the communication of motion by collision, or in any other way, have tended to confirm the supposition of this new kind of proportion. It evidently results from this, that when we have to compare forces which impress different velocities on unequal masses, each must be measured according to the product of the mass upon which it acts by the corresponding velocity. This product is called by geometers qua7itity of motion; and it determines the percussion of a body, and also the pressure that a body may exercise against any fixed obstacle to its motion.
Proceeding to the second dynamical case, we see that the characteristic difticulty of this order of questions consists in the way of estimating the connection of the different bodies of the system, in virtue of which their mutual reactions will necessarily affect the motions which each would take if alone; and we can have no a priori knowledge of what the alterations will be. In the case of the pendulum, for instance, the particles nearest the point of suspension, and those furthest from it, must react on each other by their connection, — the one moving faster and the other slower than if they had been free; and no established dynamic principle exists revealing the law which determines these reactions. Geometers naturally began by laying down a pi'inciple for each particular case; and many were the principles thus offered, which tui-ned out to be only remarkable theorems furnished simultaneously by fundamental dynamic equations. Lagrange has given us, in his " Analytical Mechanics," the general history of this series of labours: and very interesting it is, as a study of the progressive march of the human intellect. , This method of proceeding continued till mincinle ^ ^^^^ time of D'Alembert, who put an end to all these isolated researches by seeing how to compute the reactions of the liodies of a system in virtue of their connection, and establishing the fundamental ecj[uations of the motion of any system. By the aid of the great principle which bears his name, he made questions of motion merge in simple questions of equilibrium. The principle is simply this. lu the case supposed, the natural motion clearly divides itself into two, — the one which subsists and the one which has been destroyed. By D'Alembert's view, all these last, or, in other words, all the motions that have been lost or gained by the different bodies of the system by their reaction, necessarily balance each other, under the conditions of the connection which characterizes the proposed system. James Bernouilli saw this with regard to the particular case of the pendulum; and he was led by it to form an equation adapted to determine the centre of oscillation of the most simple system of weight. But he extended the resource no furtlier; and what he did detracts nothing from the credit of D'Alembert's conception, the excellence of which consists in its entire generality.